Interaction properties of complex modified Korteweg–de Vries (mKdV) solitons

Interaction properties of complex solitons are studied for the two U ( 1 ) -invariant integrable generalizations of the modified Korteweg–de Vries (mKdV) equation, given by the Hirota equation and the Sasa–Satsuma equation, which share the same traveling wave (single-soliton) solution having a sech profile characterized by a constant speed and a constant phase angle. For both equations, nonlinear interactions in which a fast soliton collides with a slow soliton are shown to be described by 2 -soliton solutions that can have three different types of interaction profile, depending on the speed ratio and the relative phase angle of the individual solitons. In all cases, the shapes and speeds of the solitons are found to be preserved apart from a shift in position such that their center of momentum moves at a constant speed. Moreover, for the Hirota equation, the phase angles of the fast and slow solitons are found to remain unchanged, while, for the Sasa–Satsuma equation, the phase angles are shown to undergo a shift such that the relative phase between the fast and slow solitons changes sign.